β-expansions of rational numbers with Pisot Chabauty basis in Qp

Abstract

The aim of this paper is to study some arithmetic properties about the periodicity of the β-expansion of p-adic numbers. We prove that for every PC unit number such that the finiteness property (F) is satisfied, there exists a constant γ 0 (β) for which every rational in the disk D(0, γ0 (β)) have a purely periodic β-expansion, where γ 0 (β) = sup{c ∈ [0, 1) : ∀x ∈ (Q ∩ Zp) ∩ [0, c), dβ(x) is purely periodic},